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National and Regional Contests
Netherlands Contests
Dutch BxMO/EGMO TST
2023 Dutch BxMO TST
1
1
Part of
2023 Dutch BxMO TST
Problems
(1)
Taking a test of n questions with increasing worth
Source: 2023 Dutch BxMO TST, Problem 1
3/12/2024
Let
n
≥
1
n \geq 1
n
≥
1
be an integer. Ruben takes a test with
n
n
n
questions. Each question on this test is worth a different number of points. The first question is worth
1
1
1
point, the second question
2
2
2
, the third
3
3
3
and so on until the last question which is worth
n
n
n
points. Each question can be answered either correctly or incorrectly. So an answer for a question can either be awarded all, or none of the points the question is worth. Let
f
(
n
)
f(n)
f
(
n
)
be the number of ways he can take the test so that the number of points awarded equals the number of questions he answered incorrectly. Do there exist in finitely many pairs
(
a
;
b
)
(a; b)
(
a
;
b
)
with
a
<
b
a < b
a
<
b
and
f
(
a
)
=
f
(
b
)
f(a) = f(b)
f
(
a
)
=
f
(
b
)
?
combinatorics